Convergence of an alternating direction and projection method for sparse dictionary learning

Sparse representation and dictionary learning have been successfully applied to encode dense data and facilitate image classification. Though existing dictionary learning methods achieve better performance than their counterparts, the class discriminative ability of learned dictionary is still limited. This paper proposes a novel supervised dictionary learning method based on the prior of the block diagonal phenomenon, i.e., each sample should be well reconstructed by the samples in the same class while poorly reconstructed by the samples in other class. Specifically, a block diagonal regularizer is imposed on the affinity matrix to enforce the sparse representation matrix to have an approximately block diagonal structure, which makes the learned dictionary more discriminative and suitable for classification tasks. Furthermore, we present an effective optimization strategy by combining the alternating minimization with the alternating direction method of multipliers (ADMM) for the proposed framework. Experimental results on six real-world datasets show that the proposed method is more effective than state-of-the-art dictionary learning methods.

Bayesian dictionary learning for hyperspectral image super resolution in mixed Poisson–Gaussian noise

Automatic prostate segmentation from MR images is an important task in various clinical applications such as prostate cancer staging and MR-guided radiotherapy planning. However, the large appearance and shape variations of the prostate in MR images make the segmentation problem difficult to solve. Traditional Active Shape/Appearance Model (ASM/AAM) has limited accuracy on this problem, since its basic assumption, i.e., both shape and appearance of the targeted organ follow Gaussian distributions, is invalid in prostate MR images. To this end, the authors propose a sparse dictionary learning method to model the image appearance in a nonparametric fashion and further integrate the appearance model into a deformable segmentation framework for prostate MR segmentation. To drive the deformable model for prostate segmentation, the authors propose nonparametric appearance and shape models. The nonparametric appearance model is based on a novel dictionary learning method, namely distributed discriminative dictionary (DDD) learning, which is able to capture fine distinctions in image appearance. To increase the differential power of traditional dictionary-based classification methods, the authors' DDD learning approach takes three strategies. First, two dictionaries for prostate and nonprostate tissues are built, respectively, using the discriminative features obtained from minimum redundancy maximum relevance feature selection. Second, linear discriminant analysis is employed as a linear classifier to boost the optimal separation between prostate and nonprostate tissues, based on the representation residuals from sparse representation. Third, to enhance the robustness of the authors' classification method, multiple local dictionaries are learned for local regions along the prostate boundary (each with small appearance variations), instead of learning one global classifier for the entire prostate. These discriminative dictionaries are located on different patches of the prostate surface and trained to adaptively capture the appearance in different prostate zones, thus achieving better local tissue differentiation. For each local region, multiple classifiers are trained based on the randomly selected samples and finally assembled by a specific fusion method. In addition to this nonparametric appearance model, a prostate shape model is learned from the shape statistics using a novel approach, sparse shape composition, which can model nonGaussian distributions of shape variation and regularize the 3D mesh deformation by constraining it within the observed shape subspace. The proposed method has been evaluated on two datasets consisting of T2-weighted MR prostate images. For the first (internal) dataset, the classification effectiveness of the authors' improved dictionary learning has been validated by comparing it with three other variants of traditional dictionary learning methods. The experimental results show that the authors' method yields a Dice Ratio of 89.1% compared to the manual segmentation, which is more accurate than the three state-of-the-art MR prostate segmentation methods under comparison. For the second dataset, the MICCAI 2012 challenge dataset, the authors' proposed method yields a Dice Ratio of 87.4%, which also achieves better segmentation accuracy than other methods under comparison. A new magnetic resonance image prostate segmentation method is proposed based on the combination of deformable model and dictionary learning methods, which achieves more accurate segmentation performance on prostate T2 MR images.

Sparse representation is efficient to approximately recover signals by a linear composition of a few bases from an over-complete dictionary. However, in the scenario of data compression, its efficiency and popularity are hindered due to the extra overhead for encoding the sparse coefficients. Therefore, how to establish an accurate rate model in sparse coding and dictionary learning becomes meaningful, which has been not fully exploited in the context of sparse representation. According to the Shannon entropy inequality, the variance of data source can bound its entropy, thus can reflect the actual coding bits. Therefore, a Globally Variance-Constrained Sparse Representation (GVCSR) model is proposed, where a variance-constrained rate term is introduced to the conventional sparse representation. To solve the non-convex optimization problem, we employ the Alternating Direction Method of Multipliers (ADMM) for sparse coding and dictionary learning, both of which have shown state-of-the-art rate-distortion performance in image representation.

Slip/antiblocking agents

By finding broader temporal and spatial patterns of brain activity, dictionary learning and sparse coding techniques circumvent the constraints of typical voxel-level studies of task-based functional magnetic resonance imaging (fMRI) data. Prior applications of these algorithms to task-related fMRI data, on the other hand, have not been tuned to simultaneously uncover temporal patterns of activity that change in tandem with changes in task circumstances and spatial patterns that exploit existing neuroscience knowledge. This work, presents a sparse dictionary learning method that decomposes fMRI data into temporal patterns of signals that loosely differ between task conditions and sparse spatial patterns that are at least partially similar to known functional network hubs, based on prior knowledge of the temporal pattern of task conditions and the locations of brain regions hypothesized to be involved in the task. To over come the a major drawback of the prior work in terms being constrained to provide an equal number of spatial and temporal patterns, even when the data was more accurately represented by a larger number spatial than temporal patterns (or vice versa), this work also propose a constrained decoupled dictionary learning method that uses prior knowledge related to task paradigms and the spatial locations of task-related brain regions to estimate decoupled spatial and temporal patterns that represent the fMRI data. The utility of sparse dictionary learning methods applied to task fMRI data is increased by simultaneously exploiting the task's known temporal structure and biasing solutions towards putative network hubs. The method identifies a high percentage of the spatial and temporal patterns programmed into simulated fMRI data. The temporal and spatial patterns are identified using an effective on-line optimization methodology. The technique detects geographical and temporal patterns in synthetic task fMRI data. When applied to two real-world fMRI data sets, the method: 1. automatically identifies temporal and spatial patterns that were known by neuroscientists to be task-related a priori but were not provided as inputs to the method; 2. provides measurements that differ significantly between differing tasks. When applied to real fMRI data from 100 participants two different studies, the suggested method also identifies geographic regions known a priori to be engaged by the Attention Network Task (ANT), AX-Continuous Performance Task (AX-CPT) and Stroop Task more thoroughly than competing methods. Using task information to partially constrain dictionary learning allowed this method to identify task related networks in real and synthetic fMRI data.

Recently, the projective dictionary pair learning (DPL) method has obtained better classification representation accuracy which learned a synthesis dictionary and an analysis dictionary to achieve the goal of signal representation. In order to obtain more discriminative ability of the dictionary pair, a new method based on DPL, called discrimination projective dictionary pair learning based dictionary learning method (DPDPL), will be proposed. In DPDPL, we will consider both the inter-class and intra-class incoherence constraints of the synthesis dictionary, and the analysis dictionary should be used to simultaneously maximize the total scatter and the between-class scatter of the signal after coding. Thus, the method ensures that the dictionary has better discriminative ability and the signals are more separable after coding. Experiments of sparse representation-based classification on several face databases show the good performances of the proposed dictionary learning method.

In this paper, we propose a simultaneous feature and dictionary learning (SFDL) method for image set based face recognition, where each training and testing example contains a face image set captured from different poses, illuminations, expressions and resolutions. While several feature learning and dictionary learning methods have been proposed for image set based face recognition in recent years, most of them learn the features and dictionaries separately, which may not be powerful enough because some discriminative information for dictionary learning may be compromised in the feature learning stage if they are applied sequentially, and vice versa. To address this, we propose a SFDL method to learn discriminative features and dictionaries simultaneously from raw face images so that discriminative information can be jointly exploited. Extensive experimental results on four widely used face datasets show that our method achieves better performance than state-of-the-art image set based face recognition methods.KeywordsFace recognitionimage setfeature learningdictionary learningsimultaneous learning

Low-rank double dictionary learning from corrupted data for robust image classification

For face recognition, conventional dictionary learning (DL) methods have disadvantages. In the paper, we propose a novel robust, discriminative and comprehensive DL (RDCDL) model. The proposed model uses sample diversities of the same face image to make the dictionary robust. The model includes class-specific dictionary atoms and disturbance dictionary atoms, which can well represent the data from different classes. Both the dictionary and the representation coefficients of data on the dictionary introduce discriminative information, which improves effectively the discrimination capability of the dictionary. The proposed RDCDL is extensively evaluated on benchmark face image databases, and it shows superior performance to many state-of-the-art sparse representation and dictionary learning methods for face recognition.

We address the problem of sparse channel representation for downlink channel estimation in multi-user frequency division duplexing (FDD) massive multiple-input multiple-output (MIMO) systems. Existing methods typically adopt discrete Fourier transform (DFT) matrix as a sparse basis to represent sparse channels. However, a sparse basis constructed through dictionary learning method has proven to have strong sparse channel representation. In this work, we develop a discriminative dictionary learning-based sparse channel representation for downlink channel estimation in a multi-user FDD massive MIMO systems. Considering partially shared support between near users, we present a new discriminative dictionary learning (DDL) method for sparse channel representation, based on which the channel estimation scheme is developed. Compared with learning a shared dictionary for all users, it can provide a better representation, thus improving the performance of recovery in the compressive sensing process. Numerical results demonstrate the superior performance of discriminative dictionary as compared to the shared dictionary in terms of normalized mean square error (NMSE) and symbol error rate (SER).

In this dissertation we develop numerical algorithms and software tools to facilitate parallel solutions of nonlinear programming (NLP) problems. In particular, we address large-scale, block-structured problems with an intrinsic decomposable configuration. These problems arise in a great number of engineering applications, including parameter estimation, optimal control, network optimization, and stochastic programming. The structure of these problems can be leveraged by optimization solvers to accelerate solutions and overcome memory limitations, and we propose variants to two classes of optimization algorithms: augmented Lagrangian (AL) schemes and Schur-complement interior-point methods. The convergence properties of augmented Lagrangian decomposition schemes like the alternating direction method of multipliers (ADMM) and progressive hedging (PH) are well established for convex optimization but convergence guarantees in non-convex settings are still poorly understood. In practice, however, ADMM and PH often perform satisfactorily in complex non-convex NLPs. In this work, we study connections between the method of multipliers (MM), ADMM, and PH to derive benchmarking metrics that explain why PH and ADMM work in practice. We illustrate the concepts using challenging dynamic optimization problems. Our exposition seeks to establish more formalism in benchmarking ADMM, PH, and AL schemes and to motivate algorithmic improvements. The effectiveness of nonlinear interior-point solvers for solving large-scale problems relies quite heavily on the solution of the underlying linear algebra systems. The schur-complement decomposition is very effective for parallelizing the solution of linear systems with modest coupling. However, for systems with large number of coupling variables the schur-complement method does not scale favorably. We implement an approach that uses a Krylov solver (GMRES) preconditioned with ADMM to solve block-structured linear systems that arise in the interior-point method. We show that this ADMM-GMRES approach overcomes the well-known scalability issues of Schur decomposition. One important drawback of using decomposition approaches like ADMM and PH is their convergence rate. Unlike Schur-complement interior-point algorithms that have super-linear convergence, augmented Lagrangian approaches typically exhibit linear and sublinear rates. We exploit connections between ADMM and the Schur-complement decomposition to derive an accelerated version of ADMM. Specifically, we study the effectiveness of performing a Newton-Raphson algorithm to compute multiplier estimates for augmented Lagrangian methods. We demonstrate using two-stage stochastic programming problems that our multiplier update achieves convergence in fewer iterations for MM on general nonlinear problems. In the case of ADMM, the newton update significantly reduces the number of subproblem solves for convex quadratic programs (QPs). Moreover, we show that using newton multiplier updates makes the method robust to the selection of the penalty parameter. Traditionally, state-of-the-art optimization solvers are implemented in low-level programming languages. In our experience, the development of decomposition algorithms in these frameworks is challenging. They present a steep learning curve and can slow the development and testing of new numerical algorithms. To mitigate these challenges, we developed PyNumero, a new open source framework implemented in Python and C++. The package seeks to facilitate development of optimization algorithms for large-scale optimization within a high-level programming environment while at the same time minimizing the computational burden of using Python. The efficiency of PyNumero is illustrated by implementing algorithms for problems arising in stochastic programming and optimal control. Timing results are presented for both serial and parallel implementations. Our computational studies demonstrate that with the appropriate balance between compiled code and Python, efficient implementations of optimization algorithms are achievable in these high-level languages.

Alternating Direction Method of Multiplier (ADMM) has been a popular algorithmic framework for separable optimization problems with linear constraints. For numerical ADMM fail to exploit the particular structure of the problem at hand nor the input data information, leveraging task-specific modules (e.g., neural networks and other data-driven architectures) to extend ADMM is a significant but challenging task. This work focuses on designing a flexible algorithmic framework to incorporate various task-specific modules (with no additional constraints) to improve the performance of ADMM in real-world applications. Specifically, we propose Guidance from Optimality (GO), a new customization strategy, to embed task-specific modules into ADMM (GO-ADMM). By introducing an optimality-based criterion to guide the propagation, GO-ADMM establishes an updating scheme agnostic to the choice of additional modules. The existing task-specific methods just plug their task-specific modules into the numerical iterations in a straightforward manner. Even with some restrictive constraints on the plug-in modules, they can only obtain some relatively weaker convergence properties for the resulted ADMM iterations. Fortunately, without any restrictions on the embedded modules, we prove the convergence of GO-ADMM regarding objective values and constraint violations, and derive the worst-case convergence rate measured by iteration complexity. Extensive experiments are conducted to verify the theoretical results and demonstrate the efficiency of GO-ADMM.

[1] Sparse geologic dictionaries provide a novel approach for subsurface flow model representation and calibration. Learning sparse dictionaries from prior training data sets is an effective approach to describe complex geologic connectivity patterns in subsurface imaging applications. However, the computational cost of sparse learning algorithms becomes prohibitive for large models. Performing the sparse dictionary learning process on smaller image patches (segments) provides a simple approach to address this problem in image processing applications. However, in underdetermined subsurface flow model calibration inverse problems, reconstruction of a segmented image can introduce significant structural distortion and discontinuity at the boundaries of the segments. This paper proposes an alternative sparse learning approach where the sparse dictionaries are learned from low-rank representations of the large-scale training data set in spectral domains (e.g., frequency domain). The objective is to develop a computationally efficient dictionary learning approach that emphasizes large-scale spatial connectivity patterns. This is achieved by removing the strong spatial correlations in the training data, thereby eliminating a large number of insignificant components from the sparse learning computation. In addition to improving the computational complexity, sparse learning from low-rank training data sets suppresses the small-scale details from entering the reconstruction of large-scale connectivity patterns, thereby providing a regularization effect in solving the resulting ill-posed inverse problems. We apply the proposed approach to travel-time tomography inversion and nonlinear subsurface flow model calibration inverse problems to demonstrate its effectiveness and practicality.

Dictionary learning has been widely used in image representation under the framework of sparse theory. But most of the current dictionary learning strategies can only be used for single-frame image separately, which are insufficient from the perspective of incremental information acquisition and global optimization for the sequential or multiframe satellite images. To this end, this paper proposes an incremental dictionary learning method for multiframe satellite images representation in the spectral domain. The incremental dictionary learning is formulated analytically in the framework of sparse representation with low-rank constraint, as a frame-by-frame gradual optimization process of global and local dictionaries, and their corresponding sparse coefficients with the sequence. Specifically, the global dictionary representing the common spectral information of the sequential frames, is optimized by two adjacent frames gradually. Meanwhile, the local dictionary representing the specific spectral information of each frame, is optimized by the newly added frame itself. In addition, an activity ratio for separating the global dictionary from the local dictionaries, an outlier detection method for initializing the local dictionary are also given, and the alternating direction methods of multipliers (ADMM) is employed to implement the above optimization. Comparison results with the related state-of-the-art methods on different datasets demonstrate that, our proposed method achieves the best representation performance in both spatial and spectral domains, and also helps to improve the performance of dictionary-based tasks using sequential satellite images, such as sea surface anomaly detection.